Show commands:
SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 39984.bj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 39984.bj1 | 39984bv6 | \([0, -1, 0, -10755385552, -429322291122752]\) | \(285531136548675601769470657/17941034271597192\) | \(8645610459214389418426368\) | \([2]\) | \(35389440\) | \(4.2452\) | |
| 39984.bj2 | 39984bv4 | \([0, -1, 0, -673490512, -6681185529920]\) | \(70108386184777836280897/552468975892674624\) | \(266229442743489645932445696\) | \([2, 2]\) | \(17694720\) | \(3.8987\) | |
| 39984.bj3 | 39984bv5 | \([0, -1, 0, -229401552, -15360815435328]\) | \(-2770540998624539614657/209924951154647363208\) | \(-101160798529098168869101535232\) | \([2]\) | \(35389440\) | \(4.2452\) | |
| 39984.bj4 | 39984bv2 | \([0, -1, 0, -71127632, 58050371520]\) | \(82582985847542515777/44772582831427584\) | \(21575473551501819207745536\) | \([2, 2]\) | \(8847360\) | \(3.5521\) | |
| 39984.bj5 | 39984bv1 | \([0, -1, 0, -55071312, 157124288448]\) | \(38331145780597164097/55468445663232\) | \(26729706143062350102528\) | \([2]\) | \(4423680\) | \(3.2055\) | \(\Gamma_0(N)\)-optimal |
| 39984.bj6 | 39984bv3 | \([0, -1, 0, 274334128, 456298688448]\) | \(4738217997934888496063/2928751705237796928\) | \(-1411337049577560353924186112\) | \([2]\) | \(17694720\) | \(3.8987\) |
Rank
sage: E.rank()
The elliptic curves in class 39984.bj have rank \(1\).
Complex multiplication
The elliptic curves in class 39984.bj do not have complex multiplication.Modular form 39984.2.a.bj
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
